Gaming Guru

You Can Prove Whatever You Want about Streaks and Trends

If you understand coin flipping, you have a good grasp on the essence of gambling,
transcending the rules of individual games. Take the idea of streaks. Or -- if
you prefer, trends, patterns, or charting -- which are ultimately variations on
a common theme. For instance, say you've watched a coin being flipped 10 times,
with the first outcome being Heads and the next nine Tails. Assuming the coin
to be unbiased and the flipper not superhumanly skilled, the reality is that the
chances of Heads or Tails on the next flip are 50 percent each. The previous flips
are irrelevant.

However, solid citizens often believe otherwise. Depending on the book or system
they bought, or on the sessions that stick out in their minds, many would argue
that the law of averages somehow drives series of events into balance so Heads
are now "due" and are therefore favored. Perhaps equally as many would
put their money on Tails, thinking the trend likely to continue. The notion
of Heads being due, "the gamblers' fallacy," is an incorrect interpretation
of probability theory. The concept of streaks continuing is, sad to say, ignorance
of the laws of the universe.

Given the original 10 flips, after another 10 trials, probability theory actually
predicts six heads and 14 Tails and not 10 each. This, because the next 10 flips
are expected to yield 5 of each. Analogously, 90 flips later, the theoretical
numbers are 46 Heads and 54 Tails, since 45 of each are expected.

The term, "expected," however is meant in a statistical sense, something
akin to "average." It doesn't connote a reasonably certain eventuality.
Still, picture what would happen were the expected numbers to materialize. The
absolute offsets would still equal eight. This was the difference, 9 - 1 = 8,
in the first 10. And, it's the expected difference, 14 - 6 = 8 or 54 - 46 =
8 after totals of 20 and 100 flips, respectively. What changes is the percentage
offset. With the 10 flips, proportions of Heads and Tails were wide apart at
1/10 and 9/10 -- 10 and 90 percent. After 20 total flips, the theoretical proportions
are 6/20 and 14/20 -- 30 and 70 percent. And when 100 flips are history, the
expected proportions are 46/100 and 54/100 or 46 and 54 percent. So values retain
the same separation but percentages converge.

To see how this might work in practice, pretend that a gambler
watched 10 flips of a coin yield one Heads and nine Tails then jumped into the
action. The accompanying table indicates what happened in a dozen different
simulated sessions after 10, 90, 990, and 9,990 additional flips. Note that
if the simulations were run again, the figures would change -- the purpose of
the exercise being to illustrate how tallies vary, rather than values actually
to be anticipated in any specific situations.

Numbers of Heads and Tails in Simulated Games of Various Lengths
after one Head and Nine Tails in the First 10 Trials

20 flips

100 flips

1,000 flips

10,000 flips

trial

Heads

Tails

Heads

Tails

Heads

Tails

Heads

Tails

1

5

15

46

54

525

475

5,006

4,994

2

8

12

57

43

504

496

4,886

5,114

3

7

13

44

56

480

520

5,107

4,983

4

7

13

44

56

475

525

4,974

5,026

5

8

12

46

54

487

513

5,041

4,959

6

5

15

53

47

502

498

4,983

5,017

7

8

12

54

46

504

496

5,037

4,963

8

5

15

36

64

493

507

4,950

5,050

9

8

12

44

56

513

487

5,030

4,970

10

5

15

49

51

505

495

4,923

5,077

11

6

14

45

55

472

528

5,002

4,998

12

5

15

42

58

497

503

5,002

4,998

The table shows that the gap between Heads and Tails can widen, narrow, or
reverse. And even in what might be considered the "long run" of 10,000
total spins, decisions don't necessarily home in precisely on the expected 4,996
and 5,004. The data in the table also suggest how easy it would be to reach
erroneous general conclusions based on what happens in particular games. For
example, after 10 additional flips, the gap between results widens in trials
1, 6, 8, and 9; this might wrongly reinforce the beliefs of those who wanted
to think that the first 10 flips showed a trend favoring Tails. On the other
hand, after 990 further flips, frequencies of the outcomes reverse in trials
1, 2, 6, 7, 9, and 10; this might be taken to falsely infer that after 10 flips
heavily biased toward Tails, the law of averages would compensate with a greater
propensity for Heads. All reminiscent of the rhyme by the punter's poet, Sumner
A Ingmark:

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